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BITSAT 2024Paper 2

Exam: BITSAT
Year: 2024
Paper: Paper 2
Subject: Mathematics
Chapter: Binomial Theorem

BITSAT 2024 · Paper 2 · Mathematics · Binomial Theorem

The coefficient of the highest power of x in the expansion of (x+...

The coefficient of the highest power of

x

in the expansion of $\left(x+\sqrt{x^2-1}\right)^8+\left(x-\sqrt{x^2-1}\right)^8

Options

A $64$
B $128$
C $256$
D $512$

Correct answer

C. $256$

Step-by-step solution

Since

\left(x+\sqrt{x^2-1}\right)^8+\left(x-\sqrt{x^2-1}\right)^8

\begin{array}{r}=2\left\{{ }^8 C_0 x^8+{ }^8 C_2 x^6\left(x^2-1\right)+{ }^8 C_4 x^4\left(x^2-1\right)^2\right. \\ \left.+{ }^8 C_6 x^2\left(x^2-1\right)^3+{ }^8 C_8 x^0\left(x^2-1\right)^4\right\}\end{array}

So coefficient of highest power of

x=2\left\{{ }^8 C_0+{ }^8 C_2+{ }^8 C_4+{ }^8 C_6+{ }^8 C_8\right\}=(1+1)^8+(1-1)^8=2^8=256

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